Problem 60 - Algebra A Combined Function

Question

In recent years, the number of newspapers printed as morning editions has been increasing and the number of newspapers printed as evening editions has been decreasing. The number \(y\) of daily morning newspapers in existence from 1997 through 2007 is approximated by the equation \(146 x-10 y=-7086\), where \(x\) is the number of years since \(1997 .\) The number \(y\) of daily evening newspapers in existence from 1997 through 2007 is approximated by \(111 x+5 y=4058,\) where \(x\) is the number of years since 1997. (Source: Based on data from Newspaper Association of America) a. Use the addition method to solve this system of equations. $$ \left\\{\begin{array}{l} 146 x-10 y=-7086 \\ 111 x+5 y=4058 \end{array}\right. $$ (Round to the nearest whole number. Because of rounding, the \(y\) -value of your ordered pair solution may vary.) b. Interpret your solution from part (a). C. How many of each type of newspaper were in existence that year?

Step-by-Step Solution

Answer

In 2000, there were 752 morning and 752 evening newspapers.

Step 1: Identify the System of Equations

We have two equations: \(146x - 10y = -7086\) and \(111x + 5y = 4058\). We will solve this system using the addition method.

Step 2: Align Equations for Elimination

To use the addition method, we need to align the coefficients of either \(x\) or \(y\) so that when we add the equations, one of the variables is eliminated. First, let's try to eliminate \(y\).

Step 3: Multiply to Align Coefficients

Multiply the first equation by 1 and the second equation by 2 to align the \(y\) coefficients:\[146x - 10y = -7086\] \[222x + 10y = 8116\]

Step 4: Add the System to Eliminate y

Add the transformed equations to eliminate \(y\):\[(146x - 10y) + (222x + 10y) = -7086 + 8116\] Simplifying the left side:\[368x = 1030\]

Step 5: Solve for x

Solve the simplified equation \(368x = 1030\) for \(x\):\[x = \frac{1030}{368} \approx 2.80\]Rounding to the nearest whole number, \(x = 3\).

Step 6: Substitute to Find y

Substitute \(x = 3\) back into one of the original equations to find \(y\). Using the first equation:\[146(3) - 10y = -7086\]\[438 - 10y = -7086\]\[-10y = -7086 - 438\]\[-10y = -7524\]\[y = \frac{-7524}{-10} = 752\]

Step 7: Solution Interpretation

The solution \((x, y)\) represents a year and the number of newspapers for that year. Therefore, \(x = 3\) implies the year 2000 (since \(1997 + 3 = 2000\)). There were 752 daily morning newspapers in 2000.

Step 8: Final Answer

In 2000, there were approximately 752 daily morning newspapers and 752 daily evening newspapers in existence.

Key Concepts

Addition Method

The addition method, also known as the elimination method, is a popular approach for solving systems of linear equations. The idea is to eliminate one of the variables by adding or subtracting the equations. This is done by adjusting the coefficients so they match for one of the variables.
For this exercise, our goal was to eliminate the variable \( y \). We first aligned the coefficients of \( y \) by observing that they were \( -10 \) and \( 5 \) in the given system. By multiplying the entire second equation by 2, the coefficients became \( 10 \) and \( -10 \), making them opposites.

• Equation 1: \ 146x - 10y = -7086
• Equation 2: \ 222x + 10y = 8116

Adding these equations canceled out \( y \), making it simple to solve for \( x \). This approach is efficient because it reduces the problem to a single-variable equation.

Solving for Variables

Once the system of equations was set up for the addition method, solving for \( x \) becomes straightforward. When the \( y \) terms were canceled out, the resulting equation was \( 368x = 1030 \).
To find \( x \), we need to divide both sides of the equation by 368:\[ x = \frac{1030}{368} \approx 2.80 \]Rounding to the nearest whole number, \( x = 3 \).
After determining \( x \), we substituted this value back into the original equations to solve for \( y \). Using Equation 1, we calculated:\[ 146(3) - 10y = -7086 \]\[-10y = -7524 \]\[ y = \frac{-7524}{-10} = 752 \]Thus, we obtained the solution \( (x, y) = (3, 752) \).

Interpreting Solutions

Interpreting the solution \( (x, y) = (3, 752) \) allows us to understand what it means in the context of the problem.
Here, \( x = 3 \) refers to the number of years since 1997, which means the year 2000. The value \( y = 752 \) represents the number of daily newspapers, both morning and evening editions, in that year.
This interpretation shows how the system of equations gives us not only a numerical solution but also real-world insights into trends over time, in this case, the circulation figures for newspapers. Recognizing how a mathematical solution corresponds to a practical situation is crucial for clear and applicable interpretations.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world phenomena. In this exercise, the equations given depict the changing numbers of morning and evening newspapers over a decade, from 1997 to 2007.
The first equation, \( 146x - 10y = -7086 \), models the trend for morning newspapers, while the second equation, \( 111x + 5y = 4058 \), represents evening newspapers. These models allow analysts to predict and compare newspaper circulations based on historical data.

• Variables \( x \) and \( y \) represent time and newspaper count, respectively
• Equations show relationships and constraints within the data

By solving these equations, one obtains specific yearly insights, reflecting how understanding mathematical models can help interpret wide-ranging data sets and forecast future events efficiently.